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\begin{document}

\title {Invertibility in Banach algebras of functional operators\\ with non-Carleman shifts}
\author{A.~Yu.~Karlovich\footnote{Instituto Superior T\'ecnico, Lisboa, Portugal.}\\
Yu.~I.~Karlovich\footnote{CINVESTAV del I.P.N., Mexico City,
Mexico.}}
\date{}
\maketitle

\begin{abstract}
\noindent We prove the inverse closedness of the Banach algebra
$\mathfrak{A}_p$ of functional operators with non-Carleman shifts,
which have only two fixed points, in the Banach algebra of all
bounded linear operators on $L^p$.\\ We suppose that $1\le
p\le\infty$ and the generators of the algebra $\mathfrak{A}_p$
have essentially bounded data. An invertibility criterion for
functional operators in $\mathfrak{A}_p$ is obtained in terms of
the invertibility of a family of discrete operators on $l^p$. An
effective invertibility criterion  is established for binomial
difference operators with $l^\infty$ coefficients on the spaces $l^p$.
Using the reduction to binomial difference operators, we give
effective criteria of invertibility  for
binomial functional operators on the spaces $L^p$.
\end{abstract}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section*{1.~Introduction}
Let $\alpha$ be an orientation-preserving homeomorphism of $[0,1]$
onto itself, which has only two fixed points $0$ and $1$. So,
$\alpha(0)=0$ and $\alpha(1)=1$, but $\alpha(t)\ne t$ for $t\in
\mathbb{I}:=(0,1)$. The function $\alpha$ is referred to as a
{\it shift}. Since the shift $\alpha$ does not satisfy the
generalized Carleman condition (see, e.g., \cite{KL94,L00}),
$\alpha$ is called a {\it non-Carleman shift}.

Denote by $\beta:=\alpha_{-1}$ the inverse function to $\alpha$.
Since $\alpha$ and $\beta$ strictly monotonically increase on
$[0,1]$, their derivatives exist and are positive almost
everywhere on $\mathbb{I}$. If $\log\alpha'\in L^\infty:=L^\infty(\mathbb{I})$,
then the shift operator $U_\alpha$ defined by
\[
(U_\alpha\varphi)(t):=(\alpha'(t))^{1/p}\varphi[\alpha(t)], \quad
t\in \mathbb{I},
\]
is an isometry on the Lebesgue space $L^p:=L^p(\mathbb{I})$ for every
$p\in[1,\infty]$. Its inverse is given by
$U_\alpha^{-1}=U_{\beta}$. Put $\alpha_0(t)=t$ and
$\alpha_n(t)=\alpha[\alpha_{n-1}(t)]$ for $n\in\mathbb{Z}$ and
$t\in[0,1]$. Then $U_\alpha^n=U_{\alpha_n}$ for $n\in\mathbb{Z}$.

Fix an arbitrary point $x\in\mathbb{I}$. Let $\gamma$ be a half-open
segment with endpoints $x$ and $\alpha(x)$ such that $x\in\gamma$
but $\alpha(x)\not\in\gamma$. Notice that either $x<\alpha(x)$ and
then $1$ is the attracting point of $\alpha$, or $\alpha(x)<x$ and
then $0$ is the attracting point of $\alpha$ (see, e.g.,
\cite[Chapter~1, Section~3]{KL94}). The shift $\alpha$ generates the
cyclic group, which is algebraically isomorphic to the group $\mathbb{Z}$
of all integer numbers. In view of this important property, we can
consider the following orbital decomposition
%%%
\begin{equation}\label{eq:orbital}
\mathbb{I}=\bigcup_{n \in \mathbb{Z}}\alpha_n(\gamma), \quad
\alpha_i(\gamma)\cap\alpha_j(\gamma)=\emptyset \quad (i\neq j).
\end{equation}
%%%

For a Banach space $X$, let $\mathfrak{B}(X)$ be the Banach
algebra of all bounded linear operators on $X$. Denote by
$\mathfrak{A}_p$ the smallest Banach subalgebra of
$\mathfrak{B}(L^p),1\le p\le\infty$, containing the operators
$U_\alpha, U_\alpha^{-1}$ and all the operators of multiplication
by functions in $L^\infty$. Thus, in what follows, always $p\in
[1,\infty]$ and the generators of $\mathfrak{A}_p$ have $L^\infty$
data. Following \cite{A, KL94}, operators $A\in \mathfrak{A}_p$
are called {\it functional operators}.

Functional operators and their discrete analogues play an
important role in the theory of functional differential operators
(see \cite{A}, \cite{Kur99} and the references therein), theory of
singular integral operators, convolution type operators and
pseudodifferential operators with shifts (see, e.g., \cite{ABL},
\cite{KL94}, \cite{L00}, \cite{Sold}), theory of dynamical systems
\cite{CL99}, etc.

The paper is devoted to several facts about the invertibility of
operators $A\in \mathfrak{A}_p$. Using the Bochner-Phillips
theorem, in Section~2 we prove the inverse closedness of
$\mathfrak{A}_p$ in $\mathfrak{B}(L^p)$. Similar results for
Wiener algebras of functional operators on Lebesgue spaces over
locally compact commutative groups, which are based on an idea of
\cite{Sem}, were established in the most general form in
\cite[Chapter~2]{Kur90} (see also \cite[Section 26]{AL} for the
inverse closedness of Wiener subalgebras in $C^*$-algebras of
abstract functional operators with discrete commutative groups of
shifts). Note also that in the case of piecewise continuous
coefficients, the inverse closedness of Banach algebras of
functional operators with discrete subexponential groups of shifts
on Lebesgue spaces over piecewise smooth contours was obtained by
other methods in \cite{K89}, \cite{KDiss}.

In Section 3, making use of a decomposition of the space $L^p$
into the direct integral of the spaces $l^p$ and generalizing the
approach of \cite[Section 26]{N64}, we get an invertibility
criterion for operators $A\in \mathfrak{A}_p$ in terms of the
invertibility of a family of discrete operators on $l^p$. In the
case of piecewise continuous coefficients analogous results for
$A\in \mathfrak{A}_p\; (1<p<\infty)$ and for functional operators
with discrete subexponential groups of shifts on the spaces $L^p\;
(1\le p\le\infty)$ were obtained in \cite{KK83}, \cite{MS80} and
in \cite{K89}, \cite{KDiss}, respectively. The latter results were
extended to $C^*$-algebras of functional operators with discrete
amenable groups of shifts in \cite{K88}, \cite{K89} (see also
\cite[Chapter 3]{A} and \cite[Section 21]{AL}).

In Section 4 we get an effective invertibility criterion for
binomial difference operators with $l^\infty$ coefficients on the
spaces $l^p$, $1\le p\le \infty$ (cf. \cite[Theorem 17.3]{AL} for
a related general $C^*$-algebra result).

Finally, in Section 5 we obtain two effective invertibility
criteria for binomial functional operators in $\mathfrak{A}_p$ on the basis
of Sections 3 and 4. Those criteria are qualitatively different
from that for binomial functional operators with data in
$C[0,1]$ (see \cite[Chapter~2, Section~4]{KL94}).

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section*{2.~Inverse closedness of $\mathfrak{A}_p$ in $\mathfrak{B}(L^p)$}
\subsection*{2.1.~The Bochner-Phillips theorem}
For a unital Banach algebra $\mathcal{B}$, let
$\mathcal{G}\mathcal{B}$ denote the group of all invertible
elements in $\mathcal{B}$. Let $\mathcal{A}$ be a Banach
subalgebra of $\mathcal{B}$ with the same identity element. The
algebra $\mathcal{A}$ is said to be {\it inverse closed} in
$\mathcal{B}$, if for every $a\in\mathcal{A}$ such that
$a\in\mathcal{G}\mathcal{B}$, we have
$a\in\mathcal{G}\mathcal{A}$.

Let $G$ be a discrete commutative group, $K=K(G)$ its character
group (all characters are continuous, and $K$ is a compact group),
and $\mathcal{L}$ a unital Banach algebra. Let $C(K,\mathcal{L})$
be the Banach algebra of all continuous functions on $K$ with
values in $\mathcal{L}$, and let $W(K,\mathcal{L})$ denote the
subalgebra of $C(K,\mathcal{L})$ which consists of the functions
\[
T:K \to\mathcal{L}, \quad \chi \mapsto \sum_{\lambda \in G} \chi (\lambda)
b_{\lambda},
\]
where $b_{\lambda}\in\mathcal{L}, \, \chi(\lambda)$ is the value
of the character $\chi$ at the element $\lambda \in G$, and
\[
\|T\|_W :=\sum_{\lambda\in G} \|b_{\lambda}\|_{\mathcal{L}} < \infty.
\]
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

%\medskip
{\bf Theorem 1.}
(see \cite[Theorem~4]{BP42} and \cite[Theorem~1.4.12]{Kur90}).
{\it The algebra $W(K,\mathcal{L})$ is inverse
closed in $C(K,\mathcal{L})$, that is, if $T(\chi)$ has an inverse in
$\mathcal{L}$ for every $\chi \in K$, then $T$ has an inverse in
$W(K,\mathcal{L})$.}
%\medskip

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection*{2.2.~Operators which commute with operators of multiplication}
The following statement is %well 
known, but for the convenience of
readers, we give its proof.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\medskip
{\bf Proposition 2.} {\it Suppose $1\le p<\infty$ and $\Delta$
is a finite segment of the real line. Every operator
$D\in\mathfrak{B}(L^p(\Delta))$ which commutes with all the
operators $\varphi I$, where $\varphi\in C(\Delta)$, has the form
$D=dI$, where $d\in L^\infty(\Delta)$.}
\medskip

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\textbf{\textit {Proof.}} By assumption, $D(gf)=gDf$ for any $g\in
C(\Delta)$ and any $f\in L^p(\Delta)$. In particular, taking
$f=1$, we get $Dg=ag$ where $a:=D(1)\in L^p(\Delta)$. Thus $D$ is
the operator of multiplication by the function $a\in L^p(\Delta)$
at least on the subset of continuous functions. Since $D\in
\mathfrak{B}(L^p(\Delta))$, one can show (assuming the contrary)
that $a\in L^\infty(\Delta)$. Hence for every $g\in C(\Delta)$,
%%%

\begin{equation} \label{eq:commute}
  \|Dg\|_{L^p(\Delta)}=\|ag\|_{L^p(\Delta)}\le
  \|a\|_{L^\infty(\Delta)}\|g\|_{L^p(\Delta)}.
\end{equation}
%%%
Since the set $C(\Delta)$ is dense in $L^p(\Delta)$, the
inequality (\ref{eq:commute}) allows us to extend $D$ by 
continuity to the whole space $L^p(\Delta)$ as the operator of
multiplication by the function $a\in L^\infty(\Delta)$.
\rule{2mm}{2mm}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection*{2.3.~Inverse closedness of the Wiener algebra of functional  operators}
Let $\mathbb{J}$ be a measurable $\alpha$-invariant (that is, 
$\alpha(\mathbb{J})=\mathbb{J}$) subset of $\mathbb{I}$. We denote by 
$\mathcal{W}_p(\mathbb{J})\; (1\le p \le\infty)$ the set of all operators
$A\in\mathfrak{B}(L^p(\mathbb{J}))$ which can be represented in the form
%%%
\begin{equation}\label{eq:Wiener-1}
A= \sum_{n \in\mathbb{Z}} a_n U_\alpha^n
\end{equation}
%%%
where $\log\alpha'\in L^\infty(\mathbb{I})$, $a_n\in L^\infty(\mathbb{J})$ and
%%%
\begin{equation}\label{eq:Wiener-2}
\|A\|_{\mathcal{W}_p(\mathbb{J})}:=\sum_{n\in\mathbb{Z}} 
\|a_n\|_{L^\infty(\mathbb{J})} < \infty.
\end{equation}
%%%
It is easy to see that $\mathcal{W}_p(\mathbb{J})$ is a Banach algebra with 
the norm $\|\cdot\|_{\mathcal{W}_p(\mathbb{J})}$. This algebra is called the 
{\it Wiener algebra} of functional operators. If $\mathbb{J}=\mathbb{I}$,
we will write $\mathcal{W}_p$ instead of $\mathcal{W}_p(\mathbb{I})$.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\medskip
{\bf Theorem 3.}
{\it The Wiener algebra $\mathcal{W}_p$ is inverse closed in $\mathfrak{B}(L^p)$.}
\medskip

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\textbf{\textit{ Proof.}} First let $1\le p<\infty$. For
$G=\mathbb{Z}$, its character group $K(G)$ coincides with the unit
circle $\mathbb{T}$, and $z(n)=z^n$ for $z \in\mathbb{T},\,
n\in\mathbb{Z}$. We apply Theorem~1 to
$\mathcal{L}=\mathfrak{B}(L^p),\, G=\mathbb{Z}$, and
$K(G)=\mathbb{T}$.

For an invertible operator $A\in\mathcal{W}_p$ given by
(\ref{eq:Wiener-1}), consider the function
%%%
\begin{equation}\label{eq:clos-1}
a:\mathbb{T}\to\mathcal{L}, \quad a(z):=\sum_{n\in\mathbb{Z}} z^n a_n U_\alpha^n.
\end{equation}
%%%
It follows from (\ref{eq:Wiener-1})--(\ref{eq:Wiener-2})  that 
$a\in W(\mathbb{T},\mathcal{L})$. For each $z\in\mathbb{T}$, let
$\varphi_z$ denote a function in $\mathcal{G} L^\infty$ which
satisfies
%%%
\begin{equation}\label{eq:clos-2}
\varphi_z [\alpha_n (t)]=z^n \varphi_z (t), \quad t \in\gamma,
\quad n \in\mathbb{Z}.
\end{equation}
%%%
Put $\Phi_z:=\varphi_z I$. In view of (\ref{eq:clos-2}),
%%%
\begin{equation}\label{eq:clos-3}
\Phi_z^{-1}U_\alpha \Phi_z=zU_\alpha, \quad z \in\mathbb{T}.
\end{equation}
%%%
For every $z \in\mathbb{T}$, we infer from (\ref{eq:clos-1}) and
(\ref{eq:clos-3}) that
%%%
\begin{equation}\label{eq:clos-4}
a(z)=\sum_{n\in\mathbb{Z}}z^n a_n U_\alpha^n
=
\Phi_z^{-1} \left(\sum_{n \in\mathbb{Z}} a_n U_\alpha^n \right)\Phi_z
=
\Phi_z^{-1} A \Phi_z.
\end{equation}
%%%
Since $A$ is invertible in $\mathcal{L}$, $a(z)$ is invertible in
$\mathcal{L}$ for every $z\in\mathbb{T}$, in view of
(\ref{eq:clos-4}). Then, due to Theorem~1, $a$ is invertible in
$W(\mathbb{T},\mathcal{L})$. Thus, its inverse has the form
%%%
\begin{equation}\label{eq:clos-5}
a^{-1}(\eta)=\sum_{n \in\mathbb{Z}} \eta^n \mathcal{D}_n, \quad \eta\in\mathbb{T},
\end{equation}
%%%
where $\mathcal{D}_n \in \mathcal{L}$ for every $n\in\mathbb{N}$,
and $\sum\limits_{n \in\mathbb{Z}} \|\mathcal{D}_n\|_\mathcal{L} < \infty.$
Since $A=a(1)$, from (\ref{eq:clos-5}) we get
%%%
\begin{equation}\label{eq:clos-6}
A^{-1}=a^{-1}(1)=\sum_{n \in\mathbb{Z}}\mathcal{D}_n.
\end{equation}
%%%

Let us  show that there exist functions $d_n \in L^{\infty}$ such
that $\mathcal{D}_n=d_n U_\alpha^n$ for all $n \in\mathbb{Z}$.
It follows from (\ref{eq:clos-4})  that
\[
\Phi_{\zeta}^{-1} a(z) \Phi_{\zeta}
=
\sum_{n\in\mathbb{Z}} \Phi_{\zeta}^{-1}\Big(z^n a_n U_\alpha^n\Big)
\Phi_{\zeta}
=
\sum_{n\in\mathbb{Z}} \zeta^n z^n a_n U_\alpha^n=a(\zeta z), \quad
z,\zeta\in\mathbb{T}.
\]
Therefore,
%%%
\begin{equation}\label{eq:clos-7}
\Phi_{\zeta}^{-1} a^{-1}(z) \Phi_{\zeta}= a^{-1}(\zeta z), \quad
z,\zeta\in\mathbb{T}.
\end{equation}
%%%
Letting $\eta=z$ and $\eta=\zeta z$ in (\ref{eq:clos-5}), we
obtain from (\ref{eq:clos-7}) that
%%%
\begin{equation}\label{eq:clos-8}
\sum_{n\in\mathbb{Z}} z^n \Phi_{\zeta}^{-1}\mathcal{D}_n \Phi_{\zeta}
=
\sum_{n\in\mathbb{Z}} z^n {\zeta}^n\mathcal{D}_n, \quad z,\zeta\in\mathbb{T}.
\end{equation}
%%%
Considering (\ref{eq:clos-8}) as a function of $z$ and
comparing the coefficients of $z^n$, we get
$\Phi_{\zeta}^{-1}\mathcal{D}_n \Phi_{\zeta}= \zeta
^n\mathcal{D}_n$ for every  $n\in\mathbb{Z}$, and thus
%%%
\begin{equation}\label{eq:clos-9}
\Phi_{\zeta}^{-1}\mathcal{D}_n \Phi_{\zeta} U_\alpha^{-n}
=
\zeta^n\mathcal{D}_n U_\alpha^{-n}, \quad n\in\mathbb{Z}.
\end{equation}
%%%
It follows from (\ref{eq:clos-3}) that
%%%
\begin{equation}\label{eq:clos-10}
\Phi_{\zeta} U_\alpha^{-n}= \zeta^n U_\alpha^{-n} \Phi_{\zeta},
\quad\zeta\in\mathbb{T},\quad n\in\mathbb{Z}.
\end{equation}
%%%
Combining (\ref{eq:clos-9})--(\ref{eq:clos-10}), we obtain
%%%
\begin{equation}\label{eq:clos-11}
\Phi_{\zeta}^{-1}\mathcal{D}_n U_\alpha^{-n} \Phi_{\zeta}
=
\mathcal{D}_n U_\alpha^{-n}, \quad n\in\mathbb{Z}.
\end{equation}
%%%
Taking into account (\ref{eq:orbital}), consider the space
$L^p=L^p(\mathbb{I})$ as the direct sum of its subspaces
$L^p(\alpha_i(\gamma)), \; i\in \mathbb{Z}$.
The operator $D_n:=\mathcal{D}_n U_\alpha^{-n} \in \mathcal{L}$
can be represented in this direct sum of subspaces
by the operator matrix
$\{\Pi_i D_n\Pi_j\}_{i,j= - \infty}^{+ \infty}$,
where $\Pi_k:=(\chi_\gamma\circ \alpha_k)I$ $(k\in\mathbb{Z})$
and $\chi_\gamma$ is the characteristic function of $\gamma$.
Since (\ref{eq:clos-11}) is valid for every function
$\varphi_{\zeta} \in \mathcal{G} L^{\infty}$ satisfying
(\ref{eq:clos-2}), we choose $\varphi_{\zeta}(t)=\zeta^i$
for $t \in \alpha_i (\gamma)$, $i\in\mathbb{Z}$; and from
(\ref{eq:clos-11}) we get
\[
\Pi_i \zeta^{-i} D_n \zeta^j \Pi_j= \Pi_i D_n \Pi_j, \quad
i,j,n\in\mathbb{Z}.
\]
Thus, $\zeta^{j-i} \Pi_i D_n \Pi_j=\Pi_i D_n \Pi_j$ whenever
$i,j,n\in\mathbb{Z}$. Choosing $\zeta \ne 1$, we get $\Pi_iD_n\Pi_j=0$ for
$i\ne j$. Hence,
%%%
\begin{equation}\label{eq:clos-12}
D_n={\rm diag}\, \{ \Pi_i D_n \Pi_i \}_{i=-\infty}^{+\infty}, \quad
n\in\mathbb{Z}.
\end{equation}
%%%

We are left with proving that each operator $\Pi_i D_n \Pi_i\;
(i,n\in\mathbb{Z})$ is an operator of multiplication by a function in
$L^{\infty}(\alpha_i(\gamma))$. Without loss of generality
assume $i=0$. It follows from (\ref{eq:clos-11}) that
$\Phi_\zeta^{-1}D_n\Phi_\zeta=D_n$, whence, in view of
(\ref{eq:clos-2}) and (\ref{eq:clos-12}), we get
%%%
\begin{equation}\label{eq:clos-13}
\varphi^{-1} \chi_\gamma D_n \chi_\gamma \varphi I=\chi_\gamma D_n
\chi_\gamma I
\end{equation}
%%%
for every $\varphi \in \mathcal{G} L^{\infty}(\gamma)$. Thus, by
(\ref{eq:clos-13}) and Proposition~2, there
exists a function $d_{n,0}\in L^\infty(\gamma)$ such that
\[
\Pi_0 D_n \Pi_0= \chi_\gamma D_n \chi_\gamma I=d_{n,0} I
\in\mathfrak{B}(L^p(\gamma)), \quad n\in\mathbb{Z}.
\]
Hence, for every $i,n\in \mathbb{Z}$, there are functions $d_{n,i}\in
L^\infty(\alpha_i(\gamma))$ for which
$
\Pi_i D_n \Pi_i=d_{n,i} I\in \mathfrak{B}(L^p(\alpha_i(\gamma)).
$
It follows from (\ref{eq:clos-12}) that $D_n=d_n I$, where
$d_n(t)=d_{n,i}(t)$ for $t\in \alpha_i(\gamma),\, i\in
\mathbb{Z}$. Finally,
%%%
\begin{equation}\label{eq:clos-14}
\mathcal{D}_n=D_n U_\alpha^n=d_n U_\alpha^n,
\quad\mbox{where}\quad d_n\in L^\infty \quad (n\in \mathbb{Z}).
\end{equation}
%%%
Combining (\ref{eq:clos-6}) and (\ref{eq:clos-14}),
we complete the proof for $p\in[1,\infty)$.

Let $p=\infty$. Obviously, for every operator $A\in\mathcal{W}_\infty$
of the form (\ref{eq:Wiener-1}), there exists the operator
\[
B=\sum_{n\in \mathbb{Z}}(\overline{a}_n\circ\alpha_{-n})U_\alpha^{-n}
\in\mathcal{W}_1
\]
such that $A=B^*$. But according to \cite[Chapter~III,
Theorem~5.30]{Kato}, the invertibility of $A$ on $L^\infty$ is
equivalent to the invertibility of $B$ on $L^1$. Thus, if $A\in
\mathcal{W}_{\infty}$ is invertible, then $B\in \mathcal{W}_1$ is 
invertible and, by the part already proved, $B^{-1}\in \mathcal{W}_1$. 
Then $A^{-1}=(B^{-1})^*\in \mathcal{W}_\infty$. 
\rule{2mm}{2mm}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\medskip
{\bf Corollary 4.} 
{\it If $\mathbb{J}\subset\mathbb{I}$ is an $\alpha$-invariant subset
of positive measure, then the Wiener algebra 
$\mathcal{W}_p(\mathbb{J})$ is inverse closed in 
$\mathfrak{B}(L^p(\mathbb{J}))$.}
\medskip

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\textbf{\textit{Proof.}} 
Setting 
$\widetilde{a}_0:=\chi_{\mathbb{J}}a_0+\chi_{\mathbb{I}\setminus\mathbb{J}}$
and $\widetilde{a}_n:=\chi_{\mathbb{J}}a_n \ (n\ne 0)$,
we get the operator
\[
\widetilde{A}:=\sum\limits_{n\in\mathbb{Z}}\widetilde{a}_n
U_\alpha^n \in \mathcal{W}_p. 
\]
Since $\mathbb{J}\subset\mathbb{I}$ is $\alpha$-invariant, 
$\widetilde{A}={\rm diag}\,\{A,I\}$ in
$\mathfrak{B}(L^p)=
\mathfrak{B}(L^p(\mathbb{J})\dot{+} L^p(\mathbb{I}\setminus\mathbb{J}))$.
Hence, $\widetilde{A}$ is invertible on $L^p$ whenever $A$ is invertible 
on $L^p(\mathbb{J})$. Applying Theorem~3 to $\widetilde{A}$, we get
$A^{-1}=(\widetilde{A})^{-1}|_{L^p(\mathbb{J})} \in \mathcal{W}_p(\mathbb{J})$. 
\rule{2mm}{2mm}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection*{2.4.~Inverse closedness of $\mathfrak{A}_p$ in $\mathfrak{B}(L^p)$}
Since the Wiener algebra $\mathcal{W}_p$ is dense in the algebra $\mathfrak{A}_p$,
from Theorem~3 we immediately get the following.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\medskip
{\bf Corollary 5.}
{\it The algebra $\mathfrak{A}_p$ is inverse closed in $\mathfrak{B}(L^p)$.}
%\medskip

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section*{3.~Relations with discrete operators}
\subsection*{3.1.~Direct integral of spaces $l^p$}
Let $1\le p\le\infty$. By analogy with \cite[Chapter~V, Section~26.5]{N64}, we say that
a vector-valued function
%%%
\begin{equation}\label{eq:vector}
f:\gamma \to l^p, \quad t\mapsto \{f_n(t)\}_{n\in \mathbb{Z}}
\end{equation}
%%%
is %called 
{\it measurable} if for an arbitrary vector $\eta\in
l^q$ $(p^{-1}+q^{-1}=1)$, the complex-valued function
\[
f_{\eta}:\gamma\to \mathbb{C}, \quad t\mapsto(f(t),\eta):=
\sum_{n\in\mathbb{Z}}f_n(t)\overline{\eta}_n
\]
is measurable on $\gamma$ with respect to the Lebesgue measure.
Since
\[
\|f(t)\|_{l^p}=\Big(\sum_{k\in\mathbb{Z}}|(f(t),e_k)|^p\Big)^{1/p},
\quad
1\le p<\infty,
\quad\quad
\|f(t)\|_{l^\infty}=
\sup_{k\in \mathbb{Z}}|(f(t),e_k)|,
\]
where $e_k\in l^q$, $(e_k)_k=1$ and $(e_k)_n=0$ $(n\ne k)$, it
follows from \cite[Chapter~I, Section~6.10, VI and VII]{N64} that
for a measurable vector-valued function (\ref{eq:vector}), the
non-negative function $t\mapsto \|f(t)\|_{l^p}$ is measurable on
$\gamma$ as well.

Further we consider the Banach space $L^p(\gamma,l^p)$ of all
measurable vector-valued functions $f:\gamma\to l^p$ with the norm
\[
\|f\|_{L^p(\gamma,l^p)}:=
\Big(\int_{\gamma}\|f(t)\|_{l^p}^p\,dt\Big)^{1/p},
\quad 1\le p<\infty,
\quad\quad
\|f\|_{L^\infty(\gamma,l^\infty)}:=
\operatornamewithlimits{ess\,sup}_{t\in \gamma}
\|f(t)\|_{l^\infty}.
\]
Thus, a function $f \in L^p(\gamma,l^p)$ is defined on $\gamma$
with the possible exclusion of a set of measure zero. For $1\le
p<\infty$, the space $L^p(\gamma,l^p)$ is called the {\it direct
integral of spaces} $l^p$.

Let $\mathcal{L}(l_p)$ be the algebra of all linear (but in
general unbounded) operators acting on $l_p$. Following
\cite[Chapter~V, Section~26.5]{N64}, an operator-valued function
$\mathcal{A}:\gamma\to\mathcal{L}(l^p)$ is said to be 
{\it measurable} if $\mathcal{A}(t)\in\mathcal{L}(l^p)$ is 
defined for all $t\in \gamma$ with the possible exclusion of 
a set of measure zero, and for an arbitrary measurable 
vector-valued function $\xi:\gamma\to l^p$, 
the vector-valued function
$\mathcal{A}\xi:\gamma\to l^p, \ t\mapsto \mathcal{A}(t)\xi(t)$
is defined for almost all $t\in\gamma$ and is measurable on
$\gamma$.

If $\mathcal{A}:\gamma\to\mathcal{L}(l^p)$ is a measurable operator-valued
function, then according to 
\cite[Chapter~V, Section~26.5, II]{N64}, we conclude that for $p\in [1,\infty)$ the
non-negative function $t\mapsto \|\mathcal{A}(\cdot)\|_{\mathfrak{B}(l^p)}$ is
measurable on $\gamma$ too. If $p=\infty$, then again the function
\[
\|\mathcal{A}(\cdot)\|_{\mathfrak{B}(l^\infty)}
= 
\sup_{i\in\mathbb{Z}}\sum_{n\in\mathbb{Z}}|(\mathcal{A}(\cdot)e_n, e_i)|
\]
is measurable on $\gamma$ as the supremum of the sequence of the
non-negative functions $\sum\limits_{n\in
\mathbb{Z}}|\mathfrak{a}_{in}(\cdot)|$ which are measurable on $\gamma$
together with $\mathfrak{a}_{in}(\cdot):=(\mathcal{A}(\cdot)e_n, e_i)$.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\medskip
{\bf Proposition 6.}
{\it A measurable operator-valued function 
$\mathcal{A}:\gamma\to\mathcal{L}(l^p), \ t\mapsto \mathcal{A}(t)$ 
defines a bounded linear operator
\[
M_\mathcal{A}:L^p(\gamma,l^p)\to L^p(\gamma,l^p), \quad (M_\mathcal{A}
f)(t)=\mathcal{A}(t)f(t), \quad t\in \gamma,
\]
if and only if the function $t\mapsto\|\mathcal{A}(t)\|_{\mathfrak{B}(l^p)}$
belongs to $L^{\infty}(\gamma)$. In that case,}
\[
\|M_\mathcal{A}\|_{\mathfrak{B}(L^p(\gamma,l^p))}
=
\Big\| \|\mathcal{A}(\cdot)\|_{\mathfrak{B}(l^p)}\Big\|_{L^{\infty}(\gamma)}.
\]
%\medskip

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\textbf{\textit{Proof.}}
Sufficiency for all $p\in [1,\infty]$ and necessity
for $p\in [1,\infty)$ are proved by analogy with
\cite[Chapter~V, Section~26.5, III]{N64}. Let us prove necessity
for $p=\infty$.

Suppose 
$M_\mathcal{A}=\mathcal{A}(\cdot)I\in\mathfrak{B}(L^\infty(\gamma,l^\infty))$. 
Then
\[
\|\mathcal{A}(t)\|_{\mathfrak{B}(l^\infty)}
=
\sup_{i\in \mathbb{Z}}\sum_{n\in\mathbb{Z}}|\mathfrak{a}_{in}(t)| 
\quad 
{\rm for}\;{\rm almost}\;{\rm all}\quad t\in\gamma.
\]
Assume that the function $t\mapsto\|\mathcal{A}(t)\|_{\mathfrak{B}(l^\infty)}$ 
does not belong to $L^\infty(\gamma)$. Then for every $m\in \mathbb{N}$ 
there is a measurable subset $\gamma_m\subset\gamma$ such that 
${\rm mes}\,\gamma_m \ne 0$ and
\[
\sup_{i\in \mathbb{Z}}\sum_{n\in \mathbb{Z}}|\mathfrak{a}_{in}(t)|>m+1 
\quad {\rm for}\;{\rm almost}\;{\rm all}\quad t\in \gamma_m.
\]
Then there are $i_m\in \mathbb{Z}$ and measurable subsets $\Delta_m\subset
\gamma_m$ such that ${\rm mes}\, \Delta_m \ne 0$ and
%%%
\begin{equation}\label{eq:nai3-1}
\|\mathcal{A}(t)\|_{\mathfrak{B}(l^\infty)} \ge
\sum_{n\in\mathbb{Z}}|\mathfrak{a}_{i_mn}(t)|\ge m 
\quad {\rm for}\;{\rm almost}\;{\rm all}\quad t\in \Delta_m.
\end{equation}
%%%
For $t\in \gamma,\, m\in \mathbb{N}$, and $n\in \mathbb{Z}$, set
%%%
\begin{equation}\label{eq:nai3-2}
f_n^{(m)}(t):=\left \{
\begin{array}{lll}
\overline{\mathfrak{a}_{i_mn}(t)}/|\mathfrak{a}_{i_mn}(t)| &
{\rm if}& \mathfrak{a}_{i_mn}(t)\ne 0,
\\[2mm]
1&{\rm if}&\mathfrak{a}_{i_mn}(t)=0.
\end{array}\right.
\end{equation}
%%%
As every function $f_n^{(m)}$ is measurable on $\gamma$ together
with $\mathfrak{a}_{i_mn}(\cdot)=(\mathcal{A}(\cdot)e_n,e_{i_m})$, the
vector functions
\[
f^{(m)}:\gamma\to l^\infty, \quad t\mapsto\{
f_n^{(m)}(t)\}_{n\in\mathbb{Z}}
\]
belong to $L^\infty(\gamma,l^\infty)$ and have norm $1$. Hence,
from (\ref{eq:nai3-1})--(\ref{eq:nai3-2}) we get for $m\in\mathbb{N}$,
%%%
\begin{eqnarray*}
&& \|\mathcal{A}\|_{\mathfrak{B}(L^\infty(\gamma,l^\infty))} \ge
\|\mathcal{A}(\cdot)f^{(m)}(\cdot)\|_{L^\infty(\gamma,l^\infty)} \ge
\operatornamewithlimits{ess\,inf}_{t\in\Delta_m}
\|\mathcal{A}(t)f^{(m)}(t)\|_{l^\infty}
\\
&& \ge \operatornamewithlimits{ess\,\inf}_{t\in \Delta_m} \left(
\sup_{i\in \mathbb{Z}}\Bigl|\sum_{n\in
\mathbb{Z}}\mathfrak{a}_{in}(t)f_n^{(m)}(t)\Bigr| \right) \ge
\operatornamewithlimits{ess\,\inf}_{t\in \Delta_m}
\Bigl|\sum_{n\in\mathbb{Z}}\mathfrak{a}_{i_mn}(t)f_n^{(m)}(t)\Bigr|
=
\operatornamewithlimits{ess\,inf}_{t\in\Delta_m} \sum_{n\in
\mathbb{Z}}|\mathfrak{a}_{i_mn}(t)|\ge m,
\end{eqnarray*}
%%%
which contradicts the boundedness of the operator $M_\mathcal{A}$ on
$L^\infty(\gamma,l^\infty)$. Thus, the function $t\mapsto
\|\mathcal{A}(t)\|_{\mathfrak{B}(l^\infty)}$ belongs to $L^\infty(\gamma)$.
\rule{2mm}{2mm}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection*{3.2.~Invertibility of functional operators in terms
of discrete operators} Below we apply the results of Section 3.1
to study the invertibility of functional operators.

Introduce the isometric isomorphism
\begin{equation}\label{f3.0}
\sigma:L^p(\mathbb{I})\to L^p(\gamma,l^p), \quad  f \mapsto \psi
\quad\; {\rm where} \quad\; \psi:\gamma\to l^p, \quad t \mapsto
\{(U_\alpha^n f)(t)\}_{n \in \mathbb{Z}}.
\end{equation}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\medskip
{\bf Lemma 7.} {\it If $A\in \mathfrak{A}_p$, then the operator
$\widehat{A}:=\sigma A\sigma^{-1}\in\mathfrak{B}(L^p(\gamma,l^p))$
is given by $(\widehat{A}\psi)(t)=\mathcal{A}(t)\psi(t)$ for almost all
$t\in\gamma$, where $\mathcal{A}$ is an operator-valued function
in $L^\infty(\gamma,\mathfrak{B}(l^p))$ which has the form}
%%%
\begin{equation}\label{f3.6}
\mathcal{A}(t)=(a_{j-i}[\alpha_i(t)])_{i,j\in \mathbb{Z}}
\quad    
for\; almost\; all
\quad t\in\gamma.
\end{equation}
%\medskip

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\textbf{\textit{Proof.}}
First suppose $A\in\mathcal{W}_p$, that is, $A$ has the form 
(\ref{eq:Wiener-1}) and satisfies (\ref{eq:Wiener-2}) with 
$\mathbb{J}=\mathbb{I}$. For $\psi\in L^p(\gamma,l^p)$
and almost all $t\in \gamma$, we get
%%%
\begin{eqnarray*}
&& (\widehat{A}\psi)(t)=(\sigma A \sigma^{-1}\psi)(t)
=
\Big\{\sum_{n\in\mathbb{Z}}a_n[\alpha_i(t)]
\Bigl(U_{\alpha}^{i+n}\sigma^{-1}\psi\Bigr)(t)\Big\}_{i\in\mathbb{Z}}
=
\Big\{\sum_{n\in\mathbb{Z}}a_n[\alpha_i(t)]\psi_{i+n}(t)\Big\}_{i\in\mathbb{Z}}
\\
&&
=
\Big\{\sum_{j\in\mathbb{Z}}a_{j-i}[\alpha_i(t)]\psi_j(t)\Big\}_{i\in\mathbb{Z}}
=
\Bigl(a_{j-i}[\alpha_i(t)]\Bigr)_{i,j\in\mathbb{Z}}\{\psi_j(t)\}_{j\in \mathbb{Z}}
=
\mathcal{A}(t)\psi(t),
\end{eqnarray*}
%%%
that is, the operator $\widehat{A}\in\mathfrak{B}(L^p(\gamma,l^p))$ 
is the operator of multiplication by the matrix function (\ref{f3.6}) 
where $a_n$ are the coefficients of the operator $A$.

Clearly, the operators $\mathcal{A}(t)$ belong to
$\mathcal{L}(l^p)$ for almost all $t\in\gamma$. Since all the
entries $a_{j-i}\circ \alpha_i$ of the matrix function
$\mathcal{A}$ belong to $L^\infty(\gamma)$, we have for almost all
$t\in\gamma$ and all $i,j\in\mathbb{Z}$,
%%%
\begin{equation}\label{eq:representation-1}
|a_{j-i}[\alpha_i(t)]|\le
\|a_{j-i}\circ\alpha_i\|_{L^\infty(\gamma)}\le\|a_{j-i}\|_{L^\infty}.
\end{equation}
%%%
It follows from (\ref{eq:representation-1}) and (\ref{eq:Wiener-2}) that 
for almost all $t\in\gamma$,
%%%
\begin{equation}\label{eq:representation-2}
\|\mathcal{A}(t)\|_{\mathfrak{B}(l^p)}
\le
\sum_{n\in\mathbb{Z}}\|a_n\|_{L^\infty}=\|A\|_{\mathcal{W}_p} <\infty.
\end{equation}
%%%
Hence, $\mathcal{A}(t)\in\mathfrak{B}(l^p)$ for those $t$.

Let us show that the operator-valued function
$\mathcal{A}:\gamma\mapsto\mathfrak{B}(l^p)$ is measurable. For 
every measurable vector-valued function $\xi:\gamma\to l^p$ and 
every $\eta\in l^q$, it follows from (\ref{eq:representation-1}) 
and H\"older's inequality  that
%%%
\begin{eqnarray}
&& \left| \sum_{n\in\mathbb{Z}}(\mathcal{A}(t)\xi(t))_n\overline{\eta}_n \right|
\le
\sum_{i\in\mathbb{Z}}\sum_{j\in\mathbb{Z}}\|a_{j-i}\|_{L^\infty}|\xi_j(t)||\eta_i|
\le
\sum_{n\in\mathbb{Z}}\|a_n\|_{L^\infty}\sum_{j\in\mathbb{Z}}|\xi_j(t)|\,|\eta_{j-n}|
\nonumber\\[3mm] &&
\le
\|A\|_{\mathcal{W}_p}\|\xi(t)\|_{l^p}\|\eta\|_{l^q} <\infty 
\quad ({\rm a.e.}\; {\rm on}\; \gamma). 
\label{eq:representation-3}
\end{eqnarray}
%%%
Since the functions $t\mapsto a_{j-i}[\alpha_i(t)]$ and
$t\mapsto\xi_j(t)\overline{\eta}_i$ are measurable on $\gamma$,
it follows  from (\ref{eq:representation-3}) and \cite[Chapter~I,
Section~6.10, VII]{N64} that the function
\[
\mathcal{A}_{\xi,\eta}:\gamma\to \mathbb{C}, \quad t\mapsto
\sum_{n\in\mathbb{Z}}(\mathcal{A}(t)\xi(t))_n\overline{\eta}_n
=
\sum_{i\in \mathbb{Z}}\sum_{j\in\mathbb{Z}}
a_{j-i}[\alpha_i(t)]\xi_j(t)\overline{\eta}_i,
\]
is well defined for almost all $t\in \gamma$ and is measurable on
$\gamma$. This means, by definition, that the operator-valued
function $\mathcal{A}$ is measurable on $\gamma$. As was shown
before, $\mathcal{A}$ defines a bounded linear operator
\[
\widehat{A}:L^p(\gamma,l^p)\to L^p(\gamma,l^p), \quad
(\widehat{A}\psi)(t)=\mathcal{A}(t)\psi(t), \quad t\in \gamma.
\]
Then due to necessity in Proposition~6, the
function $t\mapsto\|\mathcal{A}(t)\|_{\mathfrak{B}(l^p)}$ belongs to
$L^{\infty}(\gamma)$ and
\[
\Big\| 
\|\mathcal{A}(\cdot)\|_{\mathfrak{B}(l^p)}
\Big\|_{L^{\infty}(\gamma)} 
=
\|\widehat{A}\|_{\mathfrak{B}(L^p(\gamma,l^p))}.
\]

Thus, the assertion is proved for the algebra $\mathcal{W}_p$
which is dense in $\mathfrak{A}_p$. To get the assertion for the
whole algebra $\mathfrak{A}_p$ it only remains to make use of the
extension by continuity.

Indeed, if $A\in \mathfrak{A}_p\setminus \mathcal{W}_p$, then
there is a sequence $\{A_n\}\subset \mathcal{W}_p$ such that
$\lim\limits_{n\to\infty}\|A-A_n\|_{\mathfrak{B}(L^p)}=0$. In that
case the operator-valued function $\mathcal{A}:\gamma\to
\mathfrak{B}(l^p)$ is defined as the limit of the sequence of the
operator-valued functions $\mathcal{A}_n\in
L^\infty(\gamma,\mathfrak{B}(l^p))$ in
$L^\infty(\gamma,\mathfrak{B}(l^p))$. Clearly, $\mathcal{A}$ also
belongs to $L^\infty(\gamma,\mathfrak{B}(l^p))$, $\mathcal{A}$ has
the form (\ref{f3.6}), and $\mathcal{A}$ is independent of a
choice of the sequence $\{A_n\}\subset \mathcal{W}_p$. The
functions $t\mapsto a_{ij}(t):=(\mathcal{A}(t)e_j,e_i)$ are equal
to $\lim\limits_{n\to \infty}a_{ij}^{(n)}(\cdot)$ in
$L^\infty(\gamma)$, where
$a_{ij}^{(n)}(t)=(\mathcal{A}_n(t)e_j,e_i)$ are the entries of the
band-dominated (see, e.g., \cite{RRS}) operators
$\mathcal{A}_n(t)\in \mathfrak{B}(l^p)$. 
\rule{2mm}{2mm}
\medskip
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

Let $\delta_{i,j}$ stand for the Kronecker delta. From (\ref{f3.6}) 
it follows directly that the operator-valued
functions $\mathcal{A}\in L^\infty(\gamma,\mathfrak{B}(l^p))$,
associated with operators $A\in\mathfrak{A}_p$ by Lemma~7, satisfy
the following important relation evoked by (\ref{eq:orbital}).

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\medskip
{\bf Lemma 8.} {\it If $A\in\mathfrak{A}_p$, then for almost all
$t\in\gamma$ and all} $n\in\mathbb{Z}$,
\[
\mathcal{A}[\alpha_n(t)] =
\mathcal{V}^n\mathcal{A}(t)\mathcal{V}^{-n}\quad where \quad
\mathcal{V}:=(\delta_{i,j-1})_{i,j\in\mathbb{Z}}.
\]


Now we are in a position to formulate the main result of this
section.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\medskip
{\bf Theorem 9.} {\it An operator $A\in\mathfrak{A}_p$ is
invertible on $L^p$ if and only if for almost all $t\in\gamma$ the
operators $\mathcal{A}(t)$, given by Lemma~{\rm 7}, are invertible
on $l^p$ and the operator-valued function
$\mathcal{A}^{-1}:\gamma\to \mathfrak{B}(L^p)$, $t\mapsto
(\mathcal{A}(t))^{-1}$ belongs to
$L^\infty(\gamma,\mathfrak{B}(l^p))$, that is, $\mathcal{A}^{-1}$
is measurable and}
\begin{equation}\label{eq:inv-1}
\left\|\|\big(\mathcal{A}(\cdot)\big)^{-1}\|_{\mathfrak{B}(l^p)}
\right\|_{L^\infty(\gamma)} <\infty.
\end{equation}
%\medskip

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\textbf{\textit{Proof.}} {\it Necessity.} If an operator
$A\in\mathfrak{A}_p$ is invertible on $L^p$, then in view of
Corollary~5, there exists $B:=A^{-1}\in\mathfrak{A}_p$. Then the
operator $\widehat{B}:=\sigma B\sigma^{-1}\in\mathfrak{B}(L^p(\gamma,l^p))$ 
is inverse to the operator 
$\widehat{A}=\sigma A\sigma^{-1}\in\mathfrak{B}(L^p(\gamma,l^p))$. 
Since, by Lemma~7, $\widehat{A}=\mathcal{A}(\cdot)I$ and
$\widehat{B}=\mathcal{B}(\cdot)I$, where
$\mathcal{A},\mathcal{B}\in L^\infty(\gamma,\mathfrak{B}(l^p))$,
the equality $\widehat{B}=(\widehat{A})^{-1}$ implies that
$\mathcal{B}=\mathcal{A}^{-1}$. This means that for almost all
$t\in\gamma$ the operators $\mathcal{A}(t)$ are invertible on
$l^p$ and $\mathcal{A}^{-1}\in L^\infty(\gamma,
\mathfrak{B}(l^p))$. Necessity is proved.

{\it Sufficiency.} If for almost all $t\in\gamma$ the operators
$\mathcal{A}(t)$ are invertible on the space $l^p$ and (\ref{eq:inv-1})
is fulfilled, then by sufficiency of Proposition~6, the measurable
essentially bounded operator-valued function
$\mathcal{B}:\gamma\to\mathfrak{B}(l^p),\
t\mapsto\big(\mathcal{A}(t)\big)^{-1}$ generates the bounded
linear operator $\widehat{B}=
\mathcal{B}(\cdot)I\in\mathfrak{B}(L^p(\gamma,l^p))$. Since
$\mathcal{B}=\mathcal{A}^{-1}$, the operator $\widehat{B}$ is the
inverse operator for
$\widehat{A}=\mathcal{A}(\cdot)I\in\mathfrak{B}(L^p(\gamma,l^p))$.
Therefore, the operator $B=\sigma^{-1}\widehat{B}\sigma$ is the
inverse operator for $A=\sigma^{-1}\widehat{A}\sigma$.
\rule{2mm}{2mm}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section*{4.~Invertibility of binomial difference operators}
\subsection*{4.1.~Quantities characterizing invertibility}
In this section we will find criteria for the invertibility of the difference 
operator
\[
D:=aI-bV
\]
on the spaces $l^p,\, 1\le p \le\infty$, where $a,b\in l^\infty$
and the isometric shift operator $V$ is given by $(Vf)_n=f_{n+1}$
for $n\in \mathbb{Z}$. Clearly, $V$ is invertible on $l^p$, and
one can check straightforwardly the following.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\medskip
{\bf Proposition~10.} {\it For $c\in l^\infty$, the spectral radius
of operators $cV$ and $cV^{-1}$ on $l^p,1\le p\le\infty,$ 
%for every $p\in [1,\infty]$ 
is calculated by}
\[
r(c):=\lim_{n\to\infty} \left(\sup_{k\in\mathbb{Z}}|c_{k+1}c_{k+2}\ldots
c_{k+n}|\right)^{1/n}.
\]
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

%\medskip
{\bf Proposition~11.} {\it Suppose there exist $C>0$ and
$m\in\mathbb{Z}$ such that $0<C\le |c_n|<+\infty$ for
$n\in\mathbb{Z}\setminus\{m\}$.

{\rm (a)} For
\[
\rho_k^-(c):= \limsup_{n\to +\infty} |c_{k-1}c_{k-2}\dots
c_{k-n}|^{-1/n} \quad (k\le m), \quad \rho_k^+(c):= \limsup_{n\to
+\infty} |c_{k+1}c_{k+2}\dots c_{k+n}|^{1/n} \quad (k\ge m),
\]
we get
%%%
\begin{equation}\label{eq:prp2-1}
0\le\rho_k^-(c)\le \rho_s^-(c)\le C^{-1} \quad (k<s\le m),
\quad\quad C\le \rho_k^+(c)\le \rho_s^+(c)\le +\infty \quad(m\le
k<s).
\end{equation}
%%%
Moreover, there exist limits
\[
\rho_-(c):=\lim_{k\to -\infty}\rho_k^-(c)\in [0,C^{-1}], \quad
\rho_+(c):=\lim_{k\to +\infty}\rho_k^+(c)\in[C,+\infty].
\]

{\rm (b)} If $\rho_-(c)>1$, then the function $\varphi:\mathbb{Z}\to\mathbb{C}$
whose values for $n<m$ have the form
\[
\varphi_n=dc_n^{-1}c_{n+1}^{-1}\dots c_{m-1}^{-1}, \quad
d\in\mathbb{C}\setminus\{0\},
\]
does not belong to $l^p$.

{\rm (c)} If $\rho_+(c)>1$, then the function $\varphi:\mathbb{Z}\to\mathbb{C}$
whose values for $n>m$ have the form
%%%
\begin{equation}\label{eq:prp2-1a}
\varphi_n=dc_{m+1}c_{m+2}\dots c_{n-1}, \quad
d\in\mathbb{C}\setminus\{0\},
\end{equation}
%%%
does not belong to $l^p$.}
\medskip

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\textbf{\textit{Proof.}}
(a) Let us consider the ``positive'' case. Suppose $m\le k<s$, then
%%%
\begin{equation}\label{eq:prp2-2}
|c_{k+1}c_{k+2}\dots c_{k+n}|^{1/n}
=
|c_{s+1}c_{s+2}\dots c_{s+n}|^{1/n} \cdot \left(
\frac{|c_{k+1}c_{k+2}\dots c_s|}{|c_{k+n+1}c_{k+n+2}\dots
c_{s+n}|}, \right)^{1/n} \quad n\in\mathbb{N}.
\end{equation}
%%%
Further, we have
%%%
\begin{equation}\label{eq:prp2-3}
\limsup_{n\to +\infty} \left( \frac{|c_{k+1}c_{k+2}\dots
c_s|}{|c_{k+n+1}c_{k+n+2}\dots c_{s+n}|} \right)^{1/n}
\le
\limsup_{n\to +\infty} \left( \frac{|c_{k+1}c_{k+2}\dots
c_s|}{C^{s-k}} \right)^{1/n}=1.
\end{equation}
%%%
On the other hand, for every $k\ge m$ and every $n\in\mathbb{N}$, we obtain
$C\le |c_{k+1}c_{k+2}\dots c_{k+n}|^{1/n}<+\infty$.
 From the latter inequalities and (\ref{eq:prp2-2})--(\ref{eq:prp2-3})
we get the second group of inequalities in (\ref{eq:prp2-1}).
The monotonicity of the sequence $\{\rho_k^+(c)\}_{k=m}^{+\infty}$ implies
that the limit $\rho_+(c)$ exists and belongs to $[C,+\infty]$.

The ``negative'' case is considered analogously. Part (a) is
proved. Parts (b) and (c) are proved using the same idea. Let us
prove (c).

Since $\rho_+(c)>1$, by part (a) there exists a number $M\ge m$
such that $\rho_M^+(c)>1$. Therefore, there exists $q>1$ and a
sequence $n_j\to +\infty$ as $j\to +\infty$ such that
%%%
\begin{equation}\label{eq:prp2-4}
|c_Mc_{M+1}\dots c_{M+n_j}|^{1/n_j}>q>1,
\quad
j\in\mathbb{N}.
\end{equation}
%%%
Then, taking into account (\ref{eq:prp2-1a}) and (\ref{eq:prp2-4}),
we get
\[
\|\varphi\|_{l^p}\ge
|\varphi_{M+n_j+1}|>
|dc_{m+1}c_{m+2}\dots c_{M-1}|q^{n_j}
\to +\infty
\quad\mbox{as}\quad j\to +\infty
\]
whenever $d\in\mathbb{C}\setminus \{0\}$.
Proposition is proved.
\rule{2mm}{2mm}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection*{4.2.~Non-invertibility conditions}
Let the both coefficients degenerate. If $b_n\ne 0$, we put
$c_n:=a_n/b_n$ for $n\in\mathbb{Z}$.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\medskip
{\bf Proposition 12.} {\it Suppose $a,b\in l^\infty$, and  there
exist $k,m\in\mathbb{Z}$ such that
%%%
\begin{equation}\label{eq:prp3-1}
a_n\ne 0\quad\mbox{for}\quad n\in\mathbb{Z}\setminus\{k\}, \quad\quad\quad
b_n\ne 0\quad\mbox{for}\quad n\in\mathbb{Z}\setminus\{m\}, \quad\quad\quad
a_k=b_m=0.
\end{equation}
%%%
Then the operator $D$ is not invertible on $l^p$.}
\medskip

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\textbf{\textit{Proof.}} Assume that the operator $D$ is
invertible and consider the following three subcases.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{enumerate}
\item
If $m<k$ then the function $\varphi:\mathbb{Z}\to\mathbb{C}$ given by
\[
\varphi_n=\left\{
\begin{array}{ll}
0, & n\leq m,\\ 1, & n=m+1,\\ c_{m+1}c_{m+2}\dots c_{n-1}, &
m+2\leq n\leq k,\\ 0,& n>k+1
\end{array}\right.
\]
satisfies the equality $D\varphi=0$. Since  $\varphi$ has finite
support, $\varphi\in l^p$. Hence, $\varphi\in{\rm Ker}\,D\setminus\{0\}$.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\item
If $m=k$, then the non-homogeneous equation
$D\varphi=\{\delta_{m,n}\}_{n\in\mathbb{Z}}$, where $\delta_{m,n}$
is the Kronecker delta, is unsolvable on $l^p$ because, in view of
(\ref{eq:prp3-1}),
%%%
\begin{equation}\label{eq:prp3-2}
0=a_m\varphi_m-b_m\varphi_{m+1}\ne \delta_{m,m}=1.
\end{equation}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\item
If $m>k$, then the non-homogeneous equation
$D\varphi=\{\delta_{m,n}\}_{n\in\mathbb{Z}}$ is unsolvable on
$l^p$ again, because from the system
\[
a_n\varphi_n-b_n\varphi_{n+1}=0\quad (n=k,k+1, \ldots,m-1)
\]
it follows, due to $a_k=0$, that
$\varphi_{k+1}=\varphi_{k+2}=\ldots=\varphi_m=0$, and hence, we
get (\ref{eq:prp3-2}) again.
\end{enumerate}
%%%%%%%%%%%%%%%
Thus, in each subcase we get a contradiction. 
\rule{2mm}{2mm}
\medskip

Now we consider the case when the second coefficient vanishes only
at one point.

\medskip
{\bf Proposition 13.} 
{\it  Suppose $a\in\mathcal{G} l^\infty,\,
b\in l^\infty$, and there exists $m\in\mathbb{Z}$ such that $b_m=0$
and $b_n\ne 0$ for all $n\ne m$. If $\rho_+(a/b)<1$, then the
operator $D$ is not invertible on $l^p$.}
\medskip

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\textbf{\textit{Proof.}} Taking into account that $b_m=0$, $a_m\ne
0$ and $a_nb_n\ne 0$ for all $n\ne m$, we deduce that the function
$\varphi:\mathbb{Z}\to\mathbb{C}$ given by
%%%
\begin{equation}\label{eq:prp4-1}
\varphi_n=\left\{
\begin{array}{ll}
0, & n\le m,\\ 1, & n=m+1,\\ c_{m+1}c_{m+2}\dots c_{n-1}, & n>m+1
\end{array}
\right.
\end{equation}
%%%
satisfies the equation $D\varphi=0$. Since $\rho_+(a/b)<1$, from
Proposition~11(a) we get $\rho_m^+(a/b)<1$. Then, by definition,
there exist numbers $q\in (0,1)$ and $N\in\mathbb{N}$ such that
\[
|c_{m+1}c_{m+2}\dots c_{m+n}|^{1/n}<q<1 
\quad {\rm for}\; {\rm all} \quad n\ge N.
\]
Thus, $|\varphi_n|<q^{n-m}$ for $|n|\ge N$. Therefore, the
function $\varphi$ given by (\ref{eq:prp4-1}) belongs to $l^p$.
Consequently, $\varphi\in{\rm Ker}\, D\setminus \{0\}$.
This means that $D$ is not invertible. 
\rule{2mm}{2mm}
\medskip

Finally, we consider the case when the both coefficients do not
vanish on $\mathbb{Z}$.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\medskip
{\bf Proposition 14.} 
{\it Suppose $a,b\in\mathcal{G} l^\infty$. If $\rho_+(a/b)<1$ and 
$\rho_-(a/b)<1$, then the operator $D$ is not invertible on $l^p$.}
\medskip

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\textbf{\textit{Proof.}} If $\rho_-(c)<1$ and $\rho_+(c)<1$, then
there exist numbers $q\in(0,1)$ and $N\in\mathbb{N}$ such that
\[
|c_{-1}c_{-2}\dots c_{-n}|^{-1/n}<q<1, \quad |c_0c_1\dots
c_{n-1}|^{1/n}<q<1,
\]
for all $n\ge N$, respectively. Therefore, the function
$\varphi:\mathbb{Z}\to\mathbb{C}$ given by
%%%
\begin{equation}\label{eq:prp5-1}
\varphi_n=\left \{
\begin{array}{ll}
c_{n}^{-1}c_{n+1}^{-1}\ldots c_{-1}^{-1}, & n<0,\\ 1, & n=0,\\
c_0c_1\ldots c_{n-1}, & n>0
\end{array}\right.
\end{equation}
%%%
belongs to $l^p$, in view of the estimate $|\varphi_n|<q^{|n|}$ for
$|n|\ge N$. It is easily seen that $\varphi\in{\rm Ker}\, D\setminus\{0\}$. 
Thus, $D$ is not invertible. 
\rule{2mm}{2mm}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\medskip
{\bf Proposition 15.} 
{\it Suppose $a\in\mathcal{G} l^\infty,\, b\in l^\infty$, and $b_n\ne 0$ for 
all $n\in\mathbb{Z}$. If $r(b/a)>1$ and $\rho_+(a/b)>1$, then the operator 
$D$ is not invertible on $l^p$.}
\medskip

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\textbf{\textit{Proof.}} Assume that the operator $D$ is
invertible on $l^p$. Since $r(b/a)> 1$, there exist numbers $q >1$
and $M\in\mathbb{N}$ such that for every $m\ge M$ there exists
$k_m\in\mathbb{Z}$ for which
%%%
\begin{equation}\label{eq:prp6-1}
\Big|c_{k_m-1}^{-1}c_{k_m-2}^{-1}\dots c_{k_m-m}^{-1}\Big|^{1/m}
>q>1.
\end{equation}
%%%
Consider the non-homogeneous equation
$D\varphi=\{\delta_{k_m,n}\}_{n\in\mathbb{Z}}$. It is equivalent
to the system
\[
c_n\varphi_n-\varphi_{n+1}=\delta_{k_m,n}/b_n \quad
(n\in\mathbb{Z}).
\]
This equation can have only the solution 
$\varphi^{(m)}:\mathbb{Z}\to\mathbb{C}$
given by
\[
\varphi_n^{(m)}=\left\{
\begin{array}{ll}
dc_n^{-1}c_{n+1}^{-1}\dots c_{k_m-1}^{-1}, & n<k_m,\\ d, &
n=k_m,\\ dc_{k_m}-1/b_{k_m}, & n=k_m+1,\\
(dc_{k_m}-1/b_{k_m})c_{k_m+1}c_{k_m+2}\dots c_{n-1}, &n>k_m+1,
\end{array}
\right.
\]
where $d\in\mathbb{C}$. It follows from Proposition~11(c)  that
$\varphi^{(m)}$ would belong to $l^p$ only if $d=1/a_{k_m}$.

On the other hand, from (\ref{eq:prp6-1}) it follows that
\[
\|\varphi^{(m)}\|_{l^p}\ge |\varphi^{(m)}_{k_m-m}|=
\Big|a_{k_m}^{-1}c_{k_m-1}^{-1}c_{k_m-2}^{-1}\dots
c_{k_m-m}^{-1}\Big|
>Cq^m\to +\infty\quad\mbox{as}\quad m\to+\infty,
\]
where $C>0$ is the lower bound of $|a_n|^{-1}$ for
$n\in\mathbb{N}$. But this contradicts the invertibility of $D$
because for every $m\ge M$,
\[
\|\varphi^{(m)}\|_{l^p}
\le
\|D^{-1}\|_{\mathfrak{B}(l^p)}
\|\{\delta_{k_m,n}\}_{n\in\mathbb{Z}}\|_{l^p}
=
\|D^{-1}\|_{\mathfrak{B}(l^p)}<+\infty.\quad\rule{2mm}{2mm}
\]
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\medskip
{\bf Proposition 16.} {\it Suppose $a,b\in\mathcal{G} l^\infty$.
If $r(a/b)>1$ and $\rho_-(a/b)>1$, then the operator $D$ is not
invertible on $l^p$.}
\medskip

This statement is proved by analogy with Proposition~15 making use 
of Proposition~11(b).
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\subsection*{4.3.~Invertibility criterion}
Now we are in a position to prove the main result of this section.

\medskip
{\bf Theorem 17.} 
{\it Suppose $a,b\in l^\infty$. The operator
$D:=aI-bV$ is invertible on $l^p$ if and only if one of the
following two alternative conditions holds:}
%%%
\begin{equation}\label{eq:difference-1}
{\rm(i)} \quad
\inf_{n\in \mathbb{Z}}|a_n|> 0
\quad and\quad r(b/a)<1,
\quad\quad
{\rm(ii)} \quad
\inf_{n\in \mathbb{Z}}|b_n|> 0
\quad and\quad r(a/b)<1.
\end{equation}
%%%
{\it If $D$ is invertible, then its inverse is given by}
%%%
\begin{equation}\label{eq:difference-1a}
D^{-1}= \sum_{n=0}^\infty\Bigl((b/a)V\Bigr)^na^{-1}I
\quad in \; case \; {\rm (i)};
\quad\quad
D^{-1}=
-V^{-1}
\sum_{n=0}^\infty\Bigl((a/b)V^{-1}\Bigr)^nb^{-1}I
\quad
in \;case \; {\rm (ii)}.
\end{equation}
%\medskip

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\textbf{\textit{Proof.}} {\it Sufficiency}. Let $a\in\mathcal{G}
l^\infty$ and $r(b/a)<1$. Then $D=a\big(I-(b/a)V\big)$, where $b/a
\in l^{\infty}$. Since the operator $aI$ is invertible on $l^p$
and since the operator $I-(b/a)V$ also is invertible on $l^p$ in
view of the inequality $r(b/a)<1$, we infer that $D$ is invertible
on $l^p$ too, and its inverse is given by the first formula in
(\ref{eq:difference-1a}). Sufficiency of (ii) and
(\ref{eq:difference-1a}) in case (ii) are  obtained analogously.

{\it Necessity}. Assume $D$ is invertible on $l^p$ and
(\ref{eq:difference-1}) does not hold.  Then one of the following
four conditions is satisfied.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

1. Let $\inf\limits_{n\in \mathbb{Z}}|a_n|= 0,\; \inf\limits_{n\in
\mathbb{Z}}|b_n|= 0.$ Then for every $\varepsilon>0$ there exist
$\widetilde{a},\widetilde{b}\in l^\infty$ and $k,m\in\mathbb{Z}$
such that
\[
\widetilde{a}_n\ne 0\quad\mbox{for}\quad n\in\mathbb{Z}\setminus\{k\},
\quad\quad\quad \widetilde{b}_n\ne 0\quad\mbox{for}\quad
n\in\mathbb{Z}\setminus\{m\}, \quad\quad\quad
\widetilde{a}_k=\widetilde{b}_m=0,
\]
and $\|a-\widetilde{a}\|_{l^\infty}<\varepsilon/2, \
\|b-\widetilde{b}\|_{l^\infty}<\varepsilon/2$. If $\varepsilon$ is
sufficiently small, then the operator
$\widetilde{D}:=\widetilde{a}I-\widetilde{b}V$ is invertible
together with $D$ because
$\|D-\widetilde{D}\|_{\mathfrak{B}(l^p)}<\varepsilon$. But, on the
other hand, $\widetilde{D}$ is not invertible, due to
Proposition~12. So, we arrive at a contradiction.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

2. Let $\inf\limits_{n\in\mathbb{Z}}|a_n|>0, \;
\inf\limits_{n\in\mathbb{Z}}|b_n|=0, \; r(b/a) \ge 1$. Then either
$b_m=0$ for some $m\in  \mathbb{Z}$, or $b_n\ne 0$ for all $n\in
\mathbb{Z}$ but $\inf\limits_{n\in \mathbb{Z}}|b_n|=0$. Consider
these two subcases separately.

(a) If $b_m=0$ for some $m\in\mathbb{Z}$, then for $\varepsilon>0$
we define the functions
$b_{\varepsilon},\,\widetilde{b}_\varepsilon:\mathbb{Z}\to\mathbb{C}$
by
%%%
\begin{equation}\label{eq:difference-2}
(b_{\varepsilon})_n:=\left\{
\begin{array}{lll}
b_n,  & |b_n|\ge\varepsilon, & n\ne m,\\ \varepsilon, & |b_n|<\varepsilon,
 & n\ne
m,\\ 0,    & n=m,
\end{array}
\right. \quad\quad
\widetilde{b}_{\varepsilon}:=(1+\varepsilon)b_{\varepsilon}.
\end{equation}
%%%
Then, by Proposition~11(a), the quantities
$\rho_+(a/b_{\varepsilon})$ and
$\rho_+(a/\widetilde{b}_{\varepsilon})$ are well defined.
Moreover,
%%%
\begin{equation}\label{eq:difference-3}
r(\widetilde{b}_{\varepsilon}/a)=(1+\varepsilon)r(b_{\varepsilon}/a)>
r(b_{\varepsilon}/a)\ge r(b/a)\ge 1, \quad
\rho_+(a/\widetilde{b}_{\varepsilon})=(1+\varepsilon)^{-1}\rho_+(a/b_{\varepsilon}).
\end{equation}

If $\rho_+(a/b_{\varepsilon})\le 1$, then we put
$\widetilde{b}:=\widetilde{b}_{\varepsilon}$ and choose
$\varepsilon>0$ so small that the operator
$\widetilde{D}:=aI-\widetilde{b}V$ is invertible together with
$D$, and $\rho_+(a/\widetilde{b})<1$ by (\ref{eq:difference-3}).
But, on the other hand, $\widetilde{b}_m=0$ and
$\widetilde{b}_n\ne 0$ for all $n\ne m$. Hence, by Proposition~13,
the operator $\widetilde{D}$ is not invertible.

If $\rho_+(a/b_{\varepsilon})>1$, then  consider the function
$\widetilde{b}\in\mathcal{G}l^\infty$ given by
\[
\widetilde{b}_n:=\left\{
\begin{array}{ll}
(1+\varepsilon)(b_{\varepsilon})_n, & n\ne m,\\
(1+\varepsilon)\varepsilon, & n=m.
\end{array}
\right.
\]
By (\ref{eq:difference-3}), we get $r(\widetilde{b}/a)>1$.
Further, we can choose $\varepsilon>0$ so small that
$\rho_+(a/\widetilde{b})=\rho_+(a/\widetilde{b}_{\varepsilon})>1$
and the operator $\widetilde{D}:=aI-\widetilde{b}V$ is invertible
together with $D$. On the other hand, by Proposition~15, the
operator $\widetilde{D}$ is not invertible.

(b) If $b_n\ne 0$ for all $n\in\mathbb{Z}$ but
$\inf\limits_{n\in\mathbb{Z}}|b_n|=0$, then, by Proposition~11(a),
$\rho_+(a/b)$ is well defined.

If $\rho_+(a/b)>1$, then we can choose $\varepsilon>0$ so small that for
$\widetilde{b}:=(1+\varepsilon)b$ we have
%%%
\begin{equation}\label{eq:difference-6}
r(\widetilde{b}/a)=(1+\varepsilon)r(b/a)>r(b/a)\ge 1, \quad
\rho_+(a/\widetilde{b})=(1+\varepsilon)^{-1}\rho_+(a/b)>1,
\end{equation}
%%%
and the operator $\widetilde{D}:=aI-\widetilde{b}V$ is invertible
together with $D$. On the other hand, in view of Proposition~15,
the operator $\widetilde{D}$ is not invertible.

If $\rho_+(a/b)\le 1$, then for a given $\varepsilon>0$ there
exists $m\in\mathbb{Z}$ such that $|b_m|<\varepsilon$. We consider
the function $\widetilde{b}:=\widetilde{b}_{\varepsilon}$, where
$\widetilde{b}_{\varepsilon}$ is given by (\ref{eq:difference-2}).
Then, taking into account (\ref{eq:difference-3}), we have
$\rho_+(a/\widetilde{b})<1$ and $\widetilde{b}_m=0$. Clearly, we
can choose $\varepsilon>0$ so small that the operator
$\widetilde{D}:=aI-\widetilde{b}V$ is invertible together with
$D$. On the other hand, the operator $\widetilde{D}$ is not
invertible, due to Proposition~13.

Thus, case 2 is completely considered.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

3. Let $\inf\limits_{n\in\mathbb{Z}}|a_n|=0,\; \inf\limits_{n\in
\mathbb{Z}}|b_n|>0, \; r(a/b) \ge 1$. Then for the operator
$bI-aV^{-1}$, all the conditions of the previous case are satisfied.
Hence, by case 2, the operator $bI-aV^{-1}$ is not invertible. On
the other hand, the operator $bI-aV^{-1}$ is invertible on $l^p$
simultaneously with $D$.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

4. Let $\inf\limits_{n\in\mathbb{Z}}|a_n|>0, \;
\inf\limits_{n\in\mathbb{Z}}|b_n|>0, \quad r(b/a)\ge 1, \;
r(a/b)\ge 1$. In this case the characteristics $\rho_\pm(a/b)$ are
well defined, due to Proposition~11(a),  and one of the following
three conditions is fulfilled.

(a) If $\rho_+(a/b)\le 1$ and $\rho_-(a/b)\le 1$, then setting
\[
\widetilde{b}_n=\left\{
\begin{array}{ll}
(1+\varepsilon)b_n, & n\ge 0,\\ (1-\varepsilon)b_n, & n<0,
\end{array}\right.
\]
where $\varepsilon\in(0,1)$, we get
%%%
\begin{equation}\label{eq:difference-5}
\rho_+(a/\widetilde{b})=
(1+\varepsilon)^{-1}\rho_+(a/b)<\rho_+(a/b)\le 1,
\quad
\rho_-(a/\widetilde{b})=
(1-\varepsilon)\rho_-(a/b)<\rho_-(a/b)\le 1.
\end{equation}
%%%
Since
$\|b-\widetilde{b}\|_{l^\infty}\le\varepsilon\|b\|_{l^\infty}$, we
can choose $\varepsilon$ so small that the operator
$\widetilde{D}:=aI-\widetilde{b}V$ is invertible together with
$D$. But, by Proposition~14 and (\ref{eq:difference-5}), the
operator $\widetilde{D}$ is not invertible.

(b) If $\rho_+(a/b)>1$, then setting
$\widetilde{b}:=(1+\varepsilon)b$ and choosing a sufficiently
small $\varepsilon>0$, we get (\ref{eq:difference-6}), which
contradicts the invertibility of the operator
$\widetilde{D}:=aI-\widetilde{b}V$, by Proposition~15.

(c) If $\rho_-(a/b)>1$, then put $\widetilde{a}:=(1+\varepsilon)a$
where $\varepsilon>0$. Therefore, for a sufficiently small $\varepsilon>0$,
%%%
\begin{equation}\label{eq:difference-7}
r(\widetilde{a}/b)=(1+\varepsilon)r(a/b)>r(a/b)\ge 1, \quad
\rho_-(\widetilde{a}/b)=(1+\varepsilon)^{-1}\rho_-(a/b)>1.
\end{equation}
%%%
We can choose $\varepsilon$ so small that the operator
$\widetilde{D}:=\widetilde{a}I-bV$ is invertible together with
$D$. On the other hand, from (\ref{eq:difference-7}) and
Proposition~16 we deduce that the operator $\widetilde{D}$ is not
invertible. This completes the proof in case~4.

Thus, in all the cases 1--4 we get contradictions, that completes the proof. 
\rule{2mm}{2mm}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\section*{5.~Invertibility criteria for binomial functional operators}
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
In this section we get invertibility criteria on $L^p,1\le p\le\infty$,
for functional operators of the form
%%%
\begin{equation}\label{f:1}
  A=aI-bU_\alpha
\end{equation}
%%%
where $a,b, \log \alpha'\in L^\infty$. By Lemma~7, with the
operator (\ref{f:1}) we associate the operator-valued function
$\mathcal{A}\in L^\infty(\gamma,\mathfrak{B}(l^p))$ given by
\[
\mathcal{A}:t\mapsto \Big(
a[\alpha_i(t)]\delta_{i,j}-b[\alpha_i(t)]\delta_{i,j-1}
\Big)_{i,j\in\mathbb{Z}},
\]
where $\delta_{i,j}$ is the Kronecker delta. Considering vectors
in $l^p$ as complex-valued functions on $\mathbb{Z}$, we can
rewrite the operators $\mathcal{A}(t)\in \mathfrak{B}(l^p)$,
defined for almost all $t\in\gamma$, as difference operators of
the form
\[
A_t=a_t I-b_t V\in \mathfrak{B}(l^p), \quad\quad a_t:n\mapsto
a[\alpha_n(t)],\quad b_t:n\mapsto b[\alpha_n(t)]\quad (n\in \mathbb{Z}).
\]
Here $a_t,b_t$ belong to $l^\infty$ and the isometric shift
operator $V$ is given by $(Vf)_n=f_{n+1}$ for $n\in \mathbb{Z}$.

 From Theorems~9 and~17 we directly obtain the following criterion.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\medskip
{\bf Theorem 18.} {\it The operator} (\ref{f:1}) {\it is
invertible on the space $L^p$ if and only if for almost every
$t\in \gamma$ one of the following two alternative conditions
holds:}
\[
{\rm(i)} \quad
\inf_{n\in \mathbb{Z}}|a[\alpha_n(t)]|> 0
\quad and\quad r(b_t/a_t)<1,
\quad\quad
{\rm(ii)} \quad
\inf_{n\in \mathbb{Z}}|b[\alpha_n(t)]|> 0
\quad and\quad r(a_t/b_t)<1,
\]
{\it and the operator-valued function $t\mapsto
(\mathcal{A}(t))^{-1}$, where $(\mathcal{A}(t))^{-1}$ is the
matrix of the difference operator}
%%%
\begin{equation}\label{f:2}
(A_t)^{-1}:=\left\{
\begin{array}{ll}
\displaystyle \sum_{n=0}^\infty\Bigl((b_t/a_t)V\Bigr)^n(a_t)^{-1}I
& in \; case \; {\rm (i)},
\\[3mm]
-V^{-1}\displaystyle
\sum_{n=0}^\infty\Bigl((a_t/b_t)V^{-1}\Bigr)^n(b_t)^{-1}I & in \;
case \; {\rm (ii)},
\end{array}\right.
\end{equation}
%%%
{\it  belongs to $L^\infty(\gamma,\mathfrak{B}(l^p))$.}
\medskip

Making use of Theorem~18 and Corollary~4, we get the following more
pleasant criterion.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\medskip
{\bf Theorem 19.} {\it The operator} (\ref{f:1}) {\it is
invertible on the space $L^p$ if and only if there exists
partitioning of $\mathbb{I}$ into two measurable
$\alpha$-invariant subsets $\mathbb{I}_a$ and $\mathbb{I}_b$ such
that}
\[
{\rm (i)}\quad a\in \mathcal{G} L^\infty(\mathbb{I}_a), 
\quad
r\Big(((b/a)U_\alpha)|_{L^p(\mathbb{I}_a)}\Big)<1; 
\quad and \quad 
{\rm (ii)}\quad b\in \mathcal{G} L^\infty(\mathbb{I}_b),
\quad
r\Big(((a/b)U_\alpha^{-1})|_{L^p(\mathbb{I}_b)}\Big)<1.
\]

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\textbf{\textit{Proof.}} {\it Sufficiency}. 
Let ${\rm mes}\,\mathbb{I}_a>0$ and ${\rm mes}\,\mathbb{I}_b>0$. 
Taking into account that the measurable sets $\mathbb{I}_a$ and $\mathbb{I}_b$
are $\alpha$-invariant, we get
%%%
\begin{equation}\label{f:3}
A={\rm diag}\,\{A_1,A_2\}\in
\mathfrak{B}(L^p(\mathbb{I}_a)\dot{+} L^p(\mathbb{I}_b)) 
\quad
{\rm where} 
\quad  A_1:=A|_{L^p(\mathbb{I}_a)},
\quad  A_2:=A|_{L^p(\mathbb{I}_b)}.
\end{equation}
%%%
By (i) and (ii), the operator restrictions
$(I-(b/a)U_\alpha)|_{L^p(\mathbb{I}_a)}$ and
$(I-(a/b)U_\alpha^{-1})|_{L^p(\mathbb{I}_b)}$ are
invertible.~Hence,
%%%
\begin{equation}\label{f:4}
A_1^{-1}= \sum_{n=0}^\infty
\Bigl((b/a)U_\alpha\Bigr)^na^{-1}\chi_a I,\quad A_2^{-1}=
-U_\alpha^{-1}\sum_{n=0}^\infty
\Bigl((a/b)U_\alpha^{-1}\Bigr)^nb^{-1}\chi_b I
\end{equation}
%%%
where $\chi_a$ and $\chi_b$ are the characteristic functions of
$\mathbb{I}_a$ and $\mathbb{I}_b$, respectively. Finally, 
(\ref{f:3}) and (\ref{f:4}) imply that the operator $A=aI-bU_\alpha$ 
is invertible too, and
%%%
\begin{equation}\label{f:5}
A^{-1}=\sum_{n=0}^\infty\Bigl((b/a)U_\alpha\Bigr)^n a^{-1} \chi_a
I- U_\alpha^{-1}\sum_{n=0}^\infty
\Bigl((a/b)U_\alpha^{-1}\Bigr)^nb^{-1}\chi_b I
\end{equation}
%%%
Clearly, (\ref{f:5}) remains valid if ${\rm mes}\,\mathbb{I}_a =0$
or ${\rm mes}\,\mathbb{I}_b =0$. 
Sufficiency is proved.

{\it Necessity}. 
If $A$ is invertible, then for almost every $t\in\gamma$ one of 
the conditions (i)--(ii) of Theorem~18 is fulfilled. Let 
$\gamma_a$ denote the set of $t \in \gamma$ for which condition 
(i) of Theorem~18 holds, and $\gamma_b:=\gamma\setminus \gamma_a$.

Setting
\[
\mathbb{I}_a:=\bigcup_{n\in \mathbb{Z}}\alpha_n(\gamma_a), \quad
\mathbb{I}_b:=\bigcup_{n\in \mathbb{Z}}\alpha_n(\gamma_b),
\]
we infer from (\ref{eq:orbital}) that
$\mathbb{I}=\mathbb{I}_a\cup\mathbb{I}_b,\;
\mathbb{I}_a\cap\mathbb{I}_b=\emptyset$, and
$\alpha(\mathbb{I}_a)=\mathbb{I}_a$,
$\alpha(\mathbb{I}_b)=\mathbb{I}_b$. Clearly, in view of
(\ref{f:2}), the matrix of the operator $(A_t)^{-1}$ has the form
\[
(\mathcal{A}(t))^{-1}=(c_{j-i}[\alpha_i(t)])_{i,j\in \mathbb{Z}},
\quad t\in \gamma,
\]
where for almost all $t\in\gamma$,
%%%
\begin{equation}\label{f:6}
 c_n(t):=\left\{\begin{array}{ll}
\displaystyle
\frac{b(t)}{a(t)}\,\frac{b[\alpha(t)]}{a[\alpha(t)]}\ldots
\frac{b[\alpha_{n-1}(t)]}{a[\alpha_{n-1}(t)]}\,\frac{\chi_a(t)}{a[\alpha_n(t)]},
& n\in \{0,1,2,\ldots \},
\\[5mm]
\displaystyle -\frac{a[\alpha_{-1}(t)]}{b[\alpha_{-1}(t)]}\,
\frac{a[\alpha_{-2}(t)]}{b[\alpha_{-2}(t)]}\ldots
\frac{a[\alpha_{n+1}(t)]}{b[\alpha_{n+1}(t)]}\,\frac{\chi_b(t)}{b[\alpha_n(t)]},
& n\in \{-1,-2,\ldots\}.
\end{array}\right.
\end{equation}
%%%
By Lemma~8, the formulas (\ref{f:6}), defining the entries 
of $(\mathcal{A}(\cdot))^{-1}$, are valid for almost all $t\in\mathbb{I}$.

Since, by Theorem~18, the matrix function
$\mathcal{A}^{-1}:t\mapsto(\mathcal{A}(t))^{-1}$ is measurable on
$\gamma$, so is its $(0,0)$-entry
\[
c_0=(\mathcal{A}^{-1}e_0,e_0)=\chi_a/a,
\]
which implies the measurability of the sets 
$\gamma_b=\{t\in \gamma:c_0(t)=0\}$ and $\gamma_a=\gamma\setminus\gamma_b$ 
Consequently, the both sets $\mathbb{I}_a$ and $\mathbb{I}_b$ are measurable 
too. Moreover, since $\mathcal{A}^{-1}\in L^\infty(\gamma,\mathfrak{B}(l^p))$, 
we conclude that 
$c_0\circ\alpha_n=\chi_a/(a\circ\alpha_n)\in L^\infty(\gamma)$ 
for all $n\in\mathbb{Z}$. Hence, $a\in\mathcal{G} L^\infty(\mathbb{I}_a)$. 
Analogously, $b\in\mathcal{G} L^\infty(\mathbb{I}_b)$.

Let ${\rm mes}\,\gamma_a> 0$ and ${\rm mes}\,\gamma_b>0$. Since
the sets $\mathbb{I}_a$ and $\mathbb{I}_b$ are $\alpha$-invariant
and since $A$ is invertible on $L^p$, it follows from
(\ref{f:3}) that the restrictions $A_1$ and $A_2$ are invertible on 
$L^p(\mathbb{I}_a)$ and $L^p(\mathbb{I}_b)$, respectively. As
$A_1^{-1}=(\sigma^{-1}\mathcal{A}^{-1}\sigma)|_{L^p(\mathbb{I}_a)}$
and
$A_2^{-1}=(\sigma^{-1}\mathcal{A}^{-1}\sigma)|_{L^p(\mathbb{I}_b)}$,
where $\sigma$ is given by (\ref{f3.0}), we derive from
(\ref{f:6}) that, by analogy with (\ref{f:2}), the operators
$A_1^{-1}$ and $A_2^{-1}$ have the form (\ref{f:4}). 
On the other hand, by Corollary~4,
$A_1^{-1}\in\mathcal{W}_p(\mathbb{I}_a)$ and 
$A_2^{-1}\in\mathcal{W}_p(\mathbb{I}_b)$.
Thus,
\[
A_1^{-1}= \sum_{n=0}^\infty c_n U_\alpha^n 
\in\mathcal{W}_p(\mathbb{I}_a),
\quad
A_2^{-1}= -\sum_{n=1}^\infty c_{-n}U_\alpha^{-n} 
\in\mathcal{W}_p(\mathbb{I}_b),
\]
where $c_n$ are given by (\ref{f:6}) for $t\in\mathbb{I}$. Hence,
%%%
\begin{equation}\label{f:7}
\sum_{n=0}^\infty\|c_n\|_{L^\infty(\mathbb{I}_a)}<\infty, \quad
\sum_{n=1}^\infty \|c_{-n}\|_{L^\infty(\mathbb{I}_b)}<\infty.
\end{equation}
%%%
Due to the Beurling-Gelfand formula for the spectral radius,
%%%
\begin{equation}\label{f:8}
\begin{array}{l}
r_1:=
r\Big(((b/a)U_\alpha)|_{L^p(\mathbb{I}_a)}\Big)=
\lim\limits_{n\to +\infty}
\|c_n(a\circ \alpha_n)\|_{L^\infty(\mathbb{I}_a)}^{1/n}
=
\lim\limits_{n\to +\infty}\|c_n\|_{L^\infty(\mathbb{I}_a)}^{1/n},
\\[3mm]
r_2:=
r\Big(((a/b)U_\alpha^{-1})|_{L^p(\mathbb{I}_b)}\Big)=
\lim\limits_{n\to +\infty}
\|c_{-n}(b\circ\alpha_{-n})\|_{L^\infty(\mathbb{I}_b)}^{1/n}
=
\lim\limits_{n\to +\infty}
\|c_{-n}\|_{L^\infty(\mathbb{I}_b)}^{1/n}.
\end{array}
\end{equation}
%%%
By the Cauchy test for the convergence of positive series, 
from (\ref{f:7}) and (\ref{f:8}) we get $r_1\le 1$ and $r_2\le 1$.
Since the invertibility of $A$ is stable under small perturbations
of coefficients, it is easily seen that actually $r_1<1$ and $r_2<1$.

The cases ${\rm mes}\,\gamma_a =0$ and ${\rm mes}\,\gamma_b =0$
are considered analogously. 
\rule{2mm}{2mm}

\medskip
{\bf Acknowledgments.} The first author is partially supported by
F.C.T. (Portugal) grant PRAXIS
XXI/BPD/22006/99; the second author is partially supported by
CONACYT (M\'exico) Project 32726-E;
both authors are also supported by F.C.T. (Portugal) under Project
No. 34222/99--JP29.
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
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